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If x and y are each binomially distributed with index n and parameters pl and p2 respectively, then the comparison of these two binomial distributions is usually displayed as a 2 X 2 table and Fisher's "exact" (one-sided) test may be used to test the null hypothesisp1 = p2, against the alternative hypothesis that pl > p2. The exact test is based on arguing conditionally on the observed number of"successes", i.e., x + y, see Yates (1934) and Fisher (1935). The distribution of x with x + y = m fixed depends on pl and P2 only through the odds ratio 0 = (p1q2)/(q,p2), where qi 1-Pi rhe conditional distribution is Pr (x 1 0) C (n,x) C (n,y) AX/zzC(n,i)C(n,m-i)0i, where i takes the values L = max(O, m-n) to U = min(n, m). Let xc be the critical value of x for the exact test of P1 > P2 (i.e., 0 > 1 ) against the null hypothesis 0 = 1 with type I error oe, so that

For survival probabilities with censored data, Rothman (1978, Journal of Chronic Diseases 31, 557-560) has recommended the use of quadratic confidence limits based on the assumption that the product of the 'effective' sample size at time t and the life-table estimate of the survival probability past time t follows a binomial distribution. This paper shows that the proposed confidence limits are asymptotically correct for continuous survival data. These intervals, as well as those based on the arcsine transformation, the logit transformation and the log(--log) transformation, are compared by simulation to those based on Greenwood's formula--the usual method of interval estimation in life-table analysis. With large amounts of data, the alternatives to the Greenwood method all produce acceptable intervals. On the basis of overall performance, the intervals suggested by Rothman are preferred for smaller samples. Any of these methods may be used to generate confidence sets for the median survival time or for any other quantile.

Two easily-computed approximate methods of obtaining an estimate of the ratio of the hazard rates for two treatments in survival analysis are compared, by simulation, with the partial maximum likelihood (CML) estimator which assumes the Cox model with no covariates. The two methods studied are the Mantel-Haenszel estimator, and the O/E method which employs the ratio of the 'relative death rates' for the two treatments. Uncensored and censored cases are examined for both continuous and tied data with the assumption of either an underlying exponential distribution or a Weibull distribution with shape parameter 2. The O/E method is biased but, within the range of values of the ratio of the hazard rates of interest in clinical trials, it is more efficient in terms of mean square error than either CML or the Mantel-Haenszel method for all but the largest trials. The Mantel-Haenszel method is minimally biased, gives answers very close to those obtained using CML, and may be used to provide satisfactory approximate confidence intervals.

Three methods of obtaining confidence intervals for the risk ratio offailure in two independent binomial samples are compared. The three are (A) the method of Thomas and Gart (1977), (B) an adaptation of the method of Fieller using the normal distribution, and (C) a proposed method using a logarithmic transformation. On the basis of extensive simulations we have concluded that Method A is reasonable but conservative, Method B is erratic and should not be used, aszd Method C is reasonable and less conservative than Method A. Method C, cotnputationally the simplest, is recommended.